ON THE MOTION OF PENDULUMS. 71 this is a totally different thing from assuming that a motion of dilatation has no effect on the pressure at all. "When the fluid is incompressible 8 = 0, and it may be proved without difficulty that a/, G>", a>"' are constant, that is to say, constant so far as the co-ordinates are concerned. In this case we get by integrating equations (137) u = a — a)y + o)z (139). w = c —w'x + (o'y Hence, in the case of an incompressible fluid, unless the whole mass comprised within the surface S move together like a solid, there cannot fail to be a certain portion of vis viva lost by internal friction. In the case of an elastic fluid, the motion which may take place without causing a loss of vis viva in consequence of friction is somewhat more general, and corresponds to velocities u + Au, v + Av, w-\~kw, where u, v, w are the same as in (139), and AM = Sx + 2 («# + (3y + 75) x — a (a? + y* + /), with similar expressions for Av and Aw. In these expressions a, /3, 7 are three constants symmetrically related to x, y, z, and S is a constant which has the same relation to each of the co-ordinates*. 51. By means of the expression given in Art. 49, for the loss of vis viva due to internal friction, we may readily obtain a very approximate solution of the problem : To determine the rate at which the motion subsides, in consequence of internal friction, in the case of a series of oscillatory waves propagated along the surface of a liquid. Let the vertical plane of xy be parallel to the plane of motion, and let y be measured vertically downwards from the mean surface ; and for simplicity's sake suppose the depth of the fluid very great compared with the length of a wave, and the motion so small that the square of the velocity may be neglected. In the case of motion which we are considering, udx + vdy is an exact differential d<p when friction is neglected, and <£ = ce~Biysin(w#--H£) .................. (140), * See note C. at the end.